# How to calculate 𝛥𝑣 that required to change orbit with same radius but different inclination?

How could I calculate the total 𝛥𝑣 required to change the inclination of a Geostationary orbit to another circular orbit with an inclination of 30° and the same radius? (The sidereal day is used for Geostationary orbit calculation.)

Currently my idea is to solve it from the formula of inclination, $$\arccos\left(\frac{hk}{|h|}\right)$$ where $$h = rv$$ and the change from zero to 30 degrees means the component of $$h$$ on the $$k$$ axis is increasing.

Since $$r$$ does not change and $$hk$$ is changing, the velocity $$v$$ should change. I'm not too sure what to do next.

• My idea now is to solve it from the formula of inclination, arccos(hk/|h|). where h = r*v. And the change from 0 degrees to 30 degrees means the component of h on the k axis is increasing. Since r does not change, and hk is changing, the velocity v should change. I'm not too sure what to do next. Commented Nov 13, 2019 at 16:47
• en.m.wikipedia.org/wiki/Orbital_inclination_change Commented Nov 13, 2019 at 17:23
• Did you just want the equation, or were you also interested in how the equation is derived?
– uhoh
Commented Nov 13, 2019 at 17:56
• I just want the equation, thanks Commented Nov 14, 2019 at 7:14

According to Wikipedia, the general equation for inclination change is:

$$\Delta{v_i}= {2\sin(\frac{\Delta{i}}{2})\sqrt{1-e^2}\cos(\omega+f)na \over {(1+e\cos(f))}}$$

Where:

• $$e\,$$ is the orbital eccentricity
• $$\omega\,$$ is the argument of periapsis
• $$f\,$$ is the true anomaly
• $$n\,$$ is the mean motion
• $$a\,$$ is the semi-major axis

For circular orbits, this simplifies considerably to:

$$\Delta{v_i}= {2v\, \sin \left(\frac{\Delta{i}}{2} \right)}$$

Where $$v\,$$ is the orbital velocity and has the same units as $$\Delta{v_i}$$.